Formant locating system

ABSTRACT

1,034,757. Speech recognition apparatus. INTERNATIONAL BUSINESS MACHINES CORPORATION. April 5, 1965 [May 18, 1964], No. 14262/65. Heading G4R. In apparatus for determining the frequency of a resonant peak in a signal having a wide frequency band, the logarithm of the power in each of a number of frequency channels is measured, the measurements are weighted and linearly combined and the resulting signal fed to an indicator adapted to indicate the frequency of the or each peak. In the form described, the signal from source 10, Fig. 5A (not shown), is split into 36 channels by band-pass filters 20-1 to 20-36. The output of each filter passes to a square-law detector 30-1 &amp;c. to derive a signal representing the power in the corresponding channel. After smoothing at 40-1 &amp;c. the logarithm of the power signal is obtained at 50-1 &amp;c. The 36 log. power signals pass to each of 55 linear combinors 60-1 &amp;c., Fig. 5B (not shown). Each of the incoming signals is weighted by a potential divider 70-1 &amp;c., Fig. 6 (not shown), and the weightings are adjusted so that the combined output signal is a voltage proportional to the numoer of formants below a specified frequency. Each linear combinor circuit 60-1 &amp;c. has a different specified frequency. If there is one formant below the given frequency an output of 1 volt is obtained, 2 volts for 2 formants and 3 volts for 3 formants. The outputs of each combiner circuits 60-1 &amp;c. is applied to the three threshold devices 81-1 &amp;c., 82-1 &amp;c., Fig. 5C (not shown), and 83-1 &amp;c., Fig. 5D (not shown), having levels 0À5 v.,1À5 v. and 2À5 v. Those combiner circuits indicating that there is one formant below the corresponding frequency give an output from the associated threshold device 81-1 &amp;c. Those indicating two formants give an output from threshold device 82-1 &amp;c. and those indicating three formants give an output from threshold devices 83-1 &amp;c. The exact frequency of these formants is found by inverting the output from each threshold device and gating it with the output from the one above. The gates 101-1 &amp;c., 102-1 &amp;c., 103-1 respond to indicate the frequency band in which lie the first, second and third formants. Indicators 111-1 &amp;c., 112-1 &amp;c. and 113-1 &amp;c. may be printers, lamps or counters.

Feb. 13, 1968 I R. BAKIS 3,369,076

FORMANT LOCATING SYSTEM Filed May 18, 1964 '7 Sheets-Sheet 1 POWER -5 II I I I 0 LOGARITHM 0F FREQUENCY -15 I I I I I I I I -3 -2 -1 o I 2LOGARITHM OF FREQUENCY INVENTOR FIG. 2 RAIMO BAKIS ATTORNEY Feb. 13,1968 R. BAKIS 3,369,076

FORMANT LOCATING SYSTEM I Filed May 18, 1964 v 7 Sheets-Sheet 2 FIG. 3

l l l l LOGARITHM 0F FREQUENCY FIG.4

VOLTS I 1 i 1"| I I LOGARITHM 0F FREQUENCY Feb. 13, 1968 R. BAKIS3,369,076

FORMANT LOCATING SYSTEM Filed May 18, 1964 v Sheets-Sheet 5 20-i 30-140-i v K501 BP KQ' p LG F'N FILTER DETECTOR FILTER GENERATOR 20-2 40-250-2 BP aw? L P L0G F'N FlLTER DETECTOR FILTER GENERATOR 20-3 30-3 e PXW L P LOG F'N FILTER DETECTOR FILTER GENERATOR 1 4 -4 K40-4 I B P iQ'fL P LOG F'N FILTER DETECTOR F LTER GEN ERATOR w SIGNAL SOURCE /-so-35m-'35 ,50-35 B P Q w L P LOG F'N FILTER DETECTOR FILTER GENERATOR 2o-ss/30-36 4o-se ,so-3e a P w L P LOG F'N FILTER DETECTOR FILTER GENERATORFIG. FIG. FIG. FIG.

5A 5B 5C 5D Feb. 13, 1968 R BAKIS I 3,369,076

FORMANT LOCATING SYSTEM Filed May 18, 1964 FIG. 5B

7 Sheets-Sheet 4 1 00-1 01-1 91-1 101-1 111-1 I LINEAR 3 I I COMBINORCOMPARATOR I I AND INDICATOR CIRCUIT e0-2 I I 31-2 91-2 101-2 111-2LINEAR I COMBINOR COMPARATOR I I AND INDICATOR CIRCUIT T l s0-3 l 31-301-3 101-3 111-3 LINEAR 3 l COMBINOR COMPARATOR I I AND INDICATORCIRCUIT T l s0-54 I 8i54 91-54 101-54 111-54 LINEAR S I COMBINORCOMPARATOR I I AND INDICATOR CIRCUIT LINEAR 1 COMBINOR COMPARAT RCIRCuIT Feb. 13, 1968 3,369,076

R. BAKlS v FORMANT LOCATING SYSTEM Filed May 18. 1964 7 Sheets-Sheet 5 L82-1 92-I I02-I N COMPARATOR I I AND INDICATOR l 1 I I I L 82-2 92-2Io2-2 ,II2-2 I COMPARATOR I AND INDICATOR I L 82-3 92-3 Io2-5 I2-3 I I ICOMPARATOR I AND INDICATOR r I I T F L s2-s4 92-54 ,102-54 H2-54COMPARATOR I I AND INDICATOR COMPARATOR Feb. 13, 1968 F iled May 18,1964 COMPARATOR COMPARATOR COM PARATOR v R. BA'KIS FORMANT LOCATINGSYSTEM FIG.}5D I '7 Sheets-Sheet 6 INDICATOR COM PA RATOR INDICATOR IAND COMPARATOR INDICATOR INDICATOR Feb. 13, 1968 R. ,BAKIS 3,369,076

FORMANT LOCATING SYSTEM Filed May 18, 1964 '7 Sheets-Sheet 7 FROM LOGFUNCTION GENERATOR 50-1 FROM LOG FUNCTION 1 GENERATOR 50-2 FIG 6 FROMLOG FUNCTION GENERATOR 5O-5 FROM LOG FUN 1ON GENERATOR5O-5G r fi TO-5aTO-5T TO-42 H AMP AMP FREQUENCY (CYCLE PER SECOND) United States Patent3,369,076 FORMANT LOCATING SYSTEM Raimo Bakis, Ossining, N.Y.,- assignorto International Business Machines Corporation, New York, N.Y., acorporation of New York Filed May 18, 1964, Ser. No. 367,935 Claims.(CI. 179-1) ABSTRACT OF THE DISCLOSURE The power spectrum of the inputspeech signal is established and the slope of the logarithm of the powerspectrum is determined. The weighted average of the slope of thelogarithm of the power spectrum is obtained and the formant locationsare determined from the steps therein.

The present invention relates to a system for analyzing speech signalsand more particularly to a system for identifying formant frequencies.

It is accepted that an important technique in speech recognition is theidentification of formant frequencies. The relative location of at leastthe first three formant frequencies within the frequency spectrum of avoiced sound signal may be utilized to identify the sound.

In some known methods of formant identification the speech signal isapplied to a plurality of narrow-band filters and the power of thefilter output signals are measured, the output signals having higherpower values than the output signals adjacent thereto determine thepositions of the formants. A disadvantage of this method is that whentwo or more formants are positioned closely together they cannot bedistinguished.

Another known method of formant identification is by synthesis andcomparison. An initial guess is made as to the formant frequencies. Aspectrum having formants at such frequencies is synthesized and comparedwith the actual speech spectrum. This is repeated until a satisfactorymatch is produced. This method is ineflicient and time consuming becauseof the large number of trial and error calculations required.

The present invention relates to a new and improved system fordetermining the positions of formant frequencies within the frequencyspectrum of a speech signal.

Speech signals, particularly voiced sounds, are formed by bursts of airfrom the glottis passing through the vocal cavity. The vocal cavity maybe thought of as a resonant circuit. The effect of the vocal cavity onthe speech signal is similar to a number of resonant circuits in thatthe power of the output speech signal will be maximum at the frequenciesat which the vocal cavity is resonant. Referring to FIG. 1, a curve isshown which represents the logarithm of the power spectrum with respectto the logarithm of frequency for a hypothetical speech signal havingone formant. The peak shown in the curve of FIG. 1 is the result of aformant, that is, a maximum point in the power spectrum of a speechsignal. It is known that the location of the first, second and thirdformants within a speech signal may be employed to define the sound.Within limits, the first three formants in voiced sounds will repeatedlyappear at the same relative locations.

As previously stated, there are present systems for locating thepositions of formants which analyze the power spectrum of the speechsignals for the maximums. This approach fails when two or more formantsare located so close together that they appear as a single peak.

The present invention is based on a different and novel approach. Thepresent invention employs two basic functional means. A first meansresponsive to a speech signal for performing measurements thereon, saidmeasurements being linear functions of the contributions of the formantspresent in the speech signal. By linear functions of the contributionsof the formants it is meant that if there are a set of formants presentin a speech signal, and if one of them is shifted in frequency, then themeasured value changes a given amount, the given amount beingindependent of the location of the other formants. A second means isconnected to the first means for linearly combining the measurementsthereof according to a weighted average. The linearly combined signal isthen representative of the locations of the formants in the speechsignal. An indicator means may be provided which is responsive to thelinearly combined signal for displaying the frequencies at which theformants are located.

In the present embodiment the aforesaid measurements will be the slopesof the curve of the logarithm of the power spectrum of the speech signalvs. the logarithm of the frequency.

The present novel embodiment is based on the approach that the slope ofthe power spectrum after the peak value is representative of the formantlocation. This approach is based on the premise that the contribution ofindividual formants to the logarithm of the power spectrum are additiveand that for frequencies sufiiciently far below a given formant, thecontribution of that formant to the slope of the power spectrum is zero.Also, for frequencies sufiiciently far above a given formant, thecontribution is constant at a value of approximately l2 decibels peroctave.

The aforesaid can be shown by plotting the relationship of the slope ofthe power spectrum curve of FIG. 1 with respect to frequency. In FIG. 2,a curve is shown which is the slope vs. frequency relationship of thepower spectrum curve of FIG. 1. It can be seen that the location of theformant is manifested by a large peak to peak amplitude value; thatfrequencies far below the formant frequency contribute zero to thecurve; and that frequencies far above the formant frequency contribute aconstant value to the curve.

The significant aspect of the curve of FIG. 2 is that the presence of aformant causes a change in the amplitude of the curve and then, if thepeaks in the region of the formant location are ignored, the curveresembles a step function. It follows that if more than one formant wereshown in the power spectrum curve of FIG. 1, then the curve of FIG. 2would contain a separate step for each formant. As will be seen, thesystem of the present invention establishes a relation as shown in FIG.2 and measures the number and amplitude of the steps of the function todetermine the formants. As can be seen however, the amplitude of thecurve in the region of the peaks can be much greater than the amplitudeof the step. Since measurement of the amplitude of the curve in theregion 4 of the peaks would produce incorrect results, it is necessaryto eliminate the peaks. A conventional method of smoothing oreliminating the peaks in the function is to take a weighted average ofthe function, that is, to take the convolution of the curve of FIG. 2with a suit-able weighting function. It has been mathematicallydetermined by Fourier analysis that a weighting function whicheliminates the peaks in the curve of FIG. 2 is represented by thefunction shown in FIG. 3. The convolution of the weighting functionshown in FIG. 3 with the slope of the power spectrum function of FIG. 2is shown by the curve of FIG. 4. As can be seen, the peaks have beenremoved. The step function shown in FIG. 4 provides an indication of thelocation of the formant, that being the point at which the step occurs.The magnitude of the step also indicates the number of formants at thelocation. If, for example, two formants happened to be located so closetogether such as to be indistinguishable by peakdetection or or otherconventional formant locating methods, in the present system themagnitude of the step would be double, thereby indicating the presenceof two formants at approximately the same location.

The approach to locating formants in the present invention can besummarized as follows. Establish the power spectrum of the input speechsignal and determine the slope of the logarithm of the power spectrum.Obtain the weighted average of the slope of the logarithm of the powerspectrum and determine the formant locations from the steps therein. Aswill be seen, the above approach can be carried out by a practicalembodiment. In the embodiment the slope of the power spectrum isobtained by frequency separating the speech signal by narrow bandfilters, determining the power output of each filter and computing thelogarithms of the powers. The logarithms of the powers are linearlycombined according to a weighted average (convolution), the differencebetween the logarithms of the powers being representative of the slopeof the power spectrum.

An object of the present invention is to provide a system for performingmeasurements on speech signals and producing output signals whereinformants are represented by step functions.

Another object of the present invention is to provide a speechresponsive system for producing output signals having step functionsrepresenting formant frequencies and wherein the amplitudes of the stepfunctions are proportional to the number of formants at givenfrequencies.

A further object of the present invention is to provide a speechresponsive system wherein the locations of formants within a speechsignal are determined by a signal which is a combination of linearfunctions of the slope of the logarithm of the power spectrum of thespeech signal vs. the logarithm of the frequency.

Still another object of the present invention is to provide a speechresponsive system which indicates the frequencies of at least the firstthree formants of the speech signal.

The foregoing and other objects, features and advantages of theinvention will be apparent from the following more particulardescription of a preferred embodiment of the invention, as illustratedin the accompanying drawings.

In the drawings:

FIG. 1 is an illustration of the relationship between the logarithm ofthe power spectrum and the logarithm of the frequency of a hypotheticalspeech signal.

FIG. 2 is an illustration of a curve of the slope of the curveillustrated in FIG. 1.

FIG. 3 is the curve of a weighting function applied to the curve of FIG.2.

FIG. 4 is an illustration of a step function produced by applying theweighting function of FIG. 3 to the curve of FIG. 2.

FIG. 5 illustrates how FIGS. 5A, 5B, 5C and 5D should be combined.

FIGS. 5A, 5B, 5C and 5D combined represent a block diagram of a systemfor determining the positions of formant frequencies within the spectrumof a speech signal.

FIG. 6 is a detailed diagram of the linear combinor employed in thesystem of FIGS. 5A through 5D.

FIG. 7 is an illustration of step voltages obtained with the system ofFIGS. 5A through 51).

FIGS. 5A, 5B, 5C and 5D combined as shown in FIG. 5 show a block diagramof an embodiment of the present invention. A source of speech signals10, for example, a microphone provides an output analog signal of aspeech specimen. Signal source 10 is connected to a plurality of bandpass filters which, in the present embodiment are thirty-six in numberand are designated 20-1 through 20-36. Each of the filters 20-1 through20-36 have separate center frequencies and bandwidth, and in the presentexample the values are as follows:

4 TABLE I Filter Center frequency Bandwidth The filters 20-1 through20-36 will respectively pass the signals from source 10 within thedesignated frequency ranges. The outputs of the filters 20-1 through20-36 are respectively connected to square law detector circuits 30-1through 30-36. The square law detector circuits, as is well known,produce output signals which are representative of the power of theinput signals.

The outputs of the square law detector circuits 30-1 through 30-36 arerespectively connected through conventional low-pass filters 40-1through 40-36 to logarithm function generator circuits 50-1 through50-36 which provide output signals which are the logarithm function ofthe input signals thereto. The output signals from logarithm functiongenerators 50-1 through 50-36 represent the log of the power of theinput signal within the frequency range of their associated band-passfilters 20-1 through 20-36.

The outputs of each of logarithm function generators 50-1 through 50-36are connected to a plurality (fiftyfive) of linear combinor circuits60-1 through 60-55. The linear combinor circuits accept the logs of thepower signals and apply weighted averages to their differences. Each ofthe linear combinor circuits therefore has thirtysix inputs, one fromeach separate logarithm function generator. The details of linearcombinor circuit 60-1 is shown in FIG. 6. In FIG. 6 each of the inputleads from the logarithm function generators are connected to a point onparallel connected resistors 70-1 through 70-36. Each of the resistors70-1 through 70-36 have a value R. A voltage source 70-33 of +1.0 voltis connected to an R valued resistor 70-37. The relative location ofeach input lead along the length of each of the resistors ispredetermined. The location of the input lead on each resistor isdesignated by the distance the input lead is from the extreme right endof the resistor, and is referred to as j. Thus, the input lead fromlogarithm function generator 50-1 is at a position on resistor 70-1. Theinput lead from logarithm function generator 50-2 is a position onresistor 70-2 and so on. One side of the parallel combination ofresistors 70-1 through 70-37 is connected as an input to an operationalamplifier 70-39 having a feedback resistor 70-40 with a value R Theother side of the parallel combination of resistors 70-1 through 70-37is connected as an input to another operational amplifier 70-41 having afeedback resistor 70-42 also with a value R The output of amplifier70-39 is connected to the input of amplifier 70-41 through a resistor70-43 which also has a value R The output from the linear combinor 60-1is taken from the output of amplifier 70-41 on lead 71. The inputvoltages to resistors 70-1 through 70-36 from the logarithm functiongenerators are designated in FIG. as e through 2 The output voltage fromthe linear combinor circuit on lead 71 is set forth as follows:

The output voltage from a linear combinor circuit is a constant K plusthe sum of input voltages multiplied by a weighting factor y. Theweighing factor y is determined by the setting of the input lead on eachof the resistors 70-1 through 70-36. That is,

The constant K is empirically determined and compensates for suchfactors as individual characteristics of the speakers voice and otherknown distortions such as microphone response characteristics.

As previously stated, the linear combinor circuits compare and Weightthe difference between the separate power logarithms. In a simple systemthere would be only thirty-five combinor circuits provided with twoinputs each so as to respectively compare the outputs of generators 50-1and 50-2, 50-2 and 50-3, 50-3 and 50- 4, etc. In the present systemshown in FIGS. 5A through 5D the outputs of generators 50-1 and 50-2 arecompared in linear combinor 60-1, however the outputs of generators 50-3through 50-36 are also taken into account, but weighted much less thanthe outputs of generators 50-1 and 50-2. The same is true of theremainder of the linear combinor circuit. Each primarily compares theoutputs of a given two of the logarithm function generators weightedmore heavily than the other thirty-four generators. Also, fifty-fiverather than thirty-five linear combinor circuits are employed because itis possible, by judicious combining of the outputs of the nonadjacentlogarithm function generators, intermediate points could be simulated asif more than thirty-six separate frequency bands were used.

To summarize, each of the linear combinor circuits receives the outputsignals from logarithm function generator circuits 50-1 through 50-36and weights and combines them and adds a constant K to account forcharacteristics of the speakers voice. Each of the thirtysix inputsignals to each linear combinor circuit is Weighted by multiplying itwith a factor established by a variable resistor setting. The thirty-sixmultiplied signals and the constant are combined into a single outputsignal from an operational amplifier. The multiplying factor has beendesignated y and is equal to 6 where i is the resistor setting in termsof distance from the end of the resistor. In Table I a set of workablefrequency ranges were provided with relation to filters 20-1 through20-36. In the interests of a complete specification the single value ofK and the thirty-six values of y for each of the fifty-five linearcombinor circuits are set forth in Table II.

TABLE II Combinor 60-1 Logarithm generator y K Combinor -2 7 TABLEIIContinued Logarithm generator y K Combinor 60-3 Combinor 60-4 8 TABLEH-Continued Logarithm generator y K Combinor -5 Combinor 60-6 9 10 TABLEII-Continued TA E 1 o i Logarithm Logafithm generator y K generator 3 K5 50-5 -0.24679 50-35 0.01201 50-6 0.10187 50-36 -0.02052 50:] 0.0043750-8 0.01214 Combinor 60-8 s0-9 0.00917 50-10 -0.01982 504 0.040400.56579 5041 0.01449 5114 0.05096 5042 -0.03174 511-3 0.43608 50430.00066 50 1 -0.09700 5044 0.00442 505 --0.06380 5045 0.00317 50-60.17346 5046 -0.02573 50 7 0.07862 50-17 0.01078 50 s 0.00633 50-18-0.00874 50 9 0 00052 511-19 1 20 5010 -0.01292 5040 -0.00592 50111 5 250-21 000031 5042 0.03452 50-22 -0- 5043 -0.00534 50-23 0.00120 50440.00353 511-2 00467 50-15 0.00345 511-25 -0 50-16 -0.02903 50-26-0.00286 5047 0.01190 s0-27 0.00036 504 0,00959 50-28 -0- 0 8 5049-0.002S6 50-29 00047 50-20 0.00631 5030 0.00189 5041 0.00008 50-31-0.00003 5042 00534 50 2 00253 50-23 0.00108 50-33 0.00244 5 4 0 ()05 45034 0.00696 50.25 50-35 0.01157 504 0309 50-36 -0.01976 5047 )1 00()5050-2s 0.00243 Combinor 60-7 40 50-29 0.00061 5030 0.00202 50-1 0.049800.60095 50 31 0.00010 7795 50-32 0.00275 50-3 0.45719 50-33 0.00263 50 4013064 50-34 0.00756 0 1 6 -35 0.01256 50-6 0.15783 50-36 0.02147 s0 7-0.02826 50-s 0.00815 Combinor 60-9 50-9 --0.00383 50 50-10 501 0.03489-0.5245s 5041 0.01447 50 z 0.02504 5042 -0.03301 040353 50 13 -0.00275545 5044 -0.00394 50 5 -0.01647 50-15 0.00326 5 10 2 5046 0.02724 504 4.654 5047 0.01129 50-8 0.03686 50-18 0.00912 50 9 0.00347 5049 -0.0022650 10 0.00804 50-20 -0.00608 5041 1019 9 5021 0.00019 511-12 0.0358750-22 0.005 14 5q 13 ()()846 50-23 0.00113 5 00335 50-24 0.00483 5945100374 5045 -0.00055 5046 0.03113 50-26 -0.00296 511-17 0.01262 5047-0.00043 5048 0.01018 5048 0.00234 5049 --0.00287 5049 -0.00054 5040-0.00662 50-30 -0.00195 5041 0.00001 50-31 -0.00007 5022 0.00561 50420.00263 5043 0.00105 50-33 0.00253 5044 -0.00530 50-34 -0.00723 5045--0.00070 1 1 12 TABLE H-Continued TABLE H-Continued Logarithm L garithmgenerator y K generator y K 5 50-26 -0.00325 50-17 0.01440 50-27-0.00057 50-18 -0.01 171 50-28 0.00254 50-19 -0.00349 50-29 0.0006750-20 0.00751 50-30 -0.00213 50-21 -0.00014 50-31 -0.00012 50-22 0.0063650-32 -0.00289 50-23 0.00109 50-33 0.00277 50-24 -0.00602 50-34 0.0079850-25 0.00085 50-35 0.01324 50-26 -0.00369 50-36 0.02265 50-27 0.0006950-20 0.00287 Combinor 60-10 50-29 -0.00080 50-30 -0.00240 50-1 0.033170.46747 50-31 504 000 17 50-32 -0.00328 594, 0333 3 50-33 0.00313 5(} 41223 50-34 -0.00906 5g} 5 903 50-35 0.01503 5{) 6 0 13 1 50-36 0.0257350-7 O.15533 50-8 0.08541 Combinor 60-12 50-9 0.01690 50-10 -0.0052150-1 0.03381 0.32258 50-11 0.02577 50-2 0.00129 50-12 0.03651 50-30.22408 50-13 -0.01188 50-4 0.03479 50-14 0.00354 50-5 0.07452 50-150.00408 95 50-6 0.02145 50-16 0.03357 50-7 -0.00754 50-17 0.01348 50-80. 16069 50-18 0.01090 50-9 0.09440 50-19 0.00318 50-10 -0.02213 50-200.00703 50-11 0.03952 50-21 0.00008 50-12 -0.03251 50-22 -0.00596 50-13-0.01766 50-23 0.00106 50-14 0.00531 50-24 -0.00564 50-15 0.00424 50-250.00077 50-16 -0.03 865 50-26 0.00346 -17 0.01530 50-27 0.00063 50-18-0.01257 50-28 0.00269 50-19 0.00380 50-29 0.00074 50-20 0.00801 50-30-0.00226 50-21 0.00020 50-31 0.00015 50 $022 -0.00678 50-32 -0.0030850-23 0.00112 50-33 0.00294 50-24 -0.00641 50-34 -0.00849 50-25 0.0009250-35 0.01409 50-26 0.00393 50-36 -0.02411 50 50-27 0.00076 50-280.00305 Combinor -11 50-29 0.00087 50-30 0.00256 5041 0.03333 -0.39s3960 1 19 50-2 0.00223 50-32 -().0O349 3 02 943 50-33 0.00333 0 57 2 50-340.00964 001372 50-35 0.01599 gg g 5 50-30 -0.02738 50-8 -0.14207COmbinor 60-13 50-9 0.04650 50-10 -0.00854 50-1 0.03388 0.24435 231%383232 23:? 333323 50-13 -0.01521 50-4 0.01698 50-14 -0.00421 50-50.11895 50-15 0.00431 50-6 0.04381 50-16 -0.03613 7 50-7 0.02736 TABLEIICor1tinued Logarithm generator y K Combinor 60-14 14 TABLE IIContinuedCombinor 60 15 Logarithrn generator y K Combinor 60-16 1 5 TABLEII-Continued Logarithrn generator y K Combinor 60-17 Combinor 60-18 10*TABLE II-Continued Logarithm generator y K CombinOr 60-19 Combinor 60-201 7 TABLE II-Continued Logarithm generator y K Combinor 60-21 LogarithmCombinor 60-22 generator y K 504 0.02381 0.43219 50-2 0.00264 10 5030.14984 s0-4 0.05359 so-s 0.00970 50-6 0.00790 s0 7 0.01382 50-8 0.0357550 9 0.10650 5040 0.08064 5041 0.08086 5042 -0.03435 5013 0.25755 50140.00117 5045 0.03174 50-16 0.08442 5047 0.02676 5048 -0.02147 511-190.00918 5040 -0.01272 5041 0.00178 5042 -0.01071 5043 0.00065 5044-0.01015 5045 -0.0o197 5046 0.00620 5027 0.00167 5048 0.00464 5029-0.00175 -30 0.0o393 s0-31 0.00054 50 -32 0.00546 s0-33 0.00511 5044-0.01512 s0-35 0.02499 50-36 -0.04294 5 Combinor 60-23 50-1 0.022910.50145 s0-2 0.00234 s0-3 0.14258 50 4 0.05262 s0-5 0.00760 50-6 0.01343s0-7 0.01522 s0 s 0.01258 50-9 0.04407 5010 0.10080 5041 0.14502 50420.03234 5043 0.27468 5044 0.04003 5045 0.02806 50-16 0.08370 50470.02566 5048 0.02310 50 -19 --0.00992 5040 0.01338 5041 0.00209 50420.01119 5043 0.00051 5044 0.01060 5045 -0.00213 50-26 0.00646 50470.00181 5048 0.00482 19 20 TABLE IICntinued TABLE IIContinued LogarithmLogarithrn generator y K generator y K 50-29 --0.00188 5040 -0.0155550410 0.00408 50-21 --0.00284 5041 -0.00060 50-22 -0.01251 s0-32 0.00568s0-23 0.00012 s0-33 0.00530 50-24 -0.01176 50-34 --0.01575 5045 -0.00255s0-3s 0.02601 50-26 -0.00715 50-36 -0.04472 5047 -0.00217 50-28 -0.00526Combinor 60-24 50-29 -0.00220 5040 -0.00446 s0-1 0.02193 0.60316 504141-00075 50 z 000207 50-32 0.00625 50 3. 01355 50-33 0.00579 50 4 0()5174 50-34 -0.0173 3 50-5 7 51 50-35 0.02858 50.5 0 1 3 50-36 0.04919s0-7 0.01979 50-8 0.00548 Combinor 60-26 50-9 0.00523 5040 0.04518 50-10.02018 0.74879 5041 0.24275 50-2 0.00151 5042 0.03314 50-3 0.12401 50430.17800 5114 -0.04896 511-14 -0.10158 s0-5 -0.00924 50-1s -0.00176 50-60.01392 50-16 -0.07374 s0-7 0.02254 5047 0.02259 so-s 0.01324 50-18-0.02602 s0-9 -0.00282 5049 -0.01056 s0-10 0.0086s 50-20 -0.01435 50410.12400 5041 0.00246 5042 0.14415 50-22 -0.01182 5043 -0.01036 50-230.00032 5044 0.00662 5044 -0.01115 5045 -0.20510 s0-25 -0.00234 -16--0.05003 5046 -0.00680 50-17 0.02444 5047 0.00199 50-18 -0.03641 50-28-0.00503 5049 -0.01129 5049 0.00204 5040 -0.01670 50410 0.00426 s0-21-0.00330 50-31 -0.00067 5042 0.01318 50-32 --()-.00596 s0-23 -0.0001350-33 0.00554 5044 -0.01234 50414 -0.01652 50 s0-2s -0.00279 50-350.02726- 50-26 -0.00749 50-36 0.04689 50-27 -0.00237 50-28 -0.0054Combinor 60-25 0-29 50410 -0.00465 511-1 0.02102 0.69293 50-31 0.0008350-2 0.00177 5042 -0.00652 50-3 0.12930 50-33 0.00603 50-4 -0.05049s0-34 0.01810 50-5 -0.00827 5045 0.02985 50-6 0.01605 50-36 -0.0513950-7 0.02219 50 g 10073 5 Combinor 60-27 50-9 0.00651 5 s0-10 0.003965114 0.01927 0.83316 5041 022999 504 0.00120 s0-12 0.06337 s0-3 0.118245043 -0.05 931 50-4 -0.04690 50-14 -0.09925 s0-5 -0.00973 5045 -0.0896250-6 0.01130 50-16 -0.05440 50-7 0.02035 5017 0.01996 50-8 0.01662 50-18-0.03083 s0-9 0.00606 50.19 -0.01081 511-10 -0.00383 21 TABLEII-Continued Logarithm generator y K Combinor 60-28 22 TABLE IIContinuedCombinor 60-29 Logarithm generator y K Combinor 60-30 23 24 TABLEII-Continued TABLE IIContinued Logarithm Logarithm generator K generatory K 5 50-30 0.00547 50-21 0.00789 50-31 0.00131 50-22 0.01970 50-320.00777 50-23 0.00292 50-33 0.00706 50-24 0.01728 50-34 0.02160 50-250.00493 50-35 0.03549 50-26 0.01026 50-36 0.06 130 50-27 0.00413 50-280.00712 Combinor 60-31 50-29 0.00389 50-30 0.00607 5114 0.01607 1.0809511-31 000156 50.2 090050 50-32 0.00861 50 3 1097 5 50-33 0.00775 50 43954 50-34 0.02391 50-5 0.00865 035 0.03923 50 6 090309 50-36 0.0678250-7 0.01425 50-8 001089 Combinor 60-33 50-9 0.00560 50-10 0.00233 5040.01482 1.2536 50-11 0.04281 50-2 0.00004 50-12 0.04009 50-3 0.0894250-13 0.06624 50-4 0.03684 50-14 0.08044 50-5 0.00840 50-15 0.30244 so-s0.00693 50-16 0.17 876 50-7 0.01266 50-17 -0.12120 50-8 0.00978 50-180.02061 50-9 0.00471 50-19 0.03391 50-10 0.00092 50-20 -0.02161 50-110.03162 50-21 -0.00855 50-12 0.04196 50-22 0.01745 50-13 0.04050 50-230.00250 50-14 0.02578 50-24 0.01607 40 50-15 0.15876 50-25 0.00450 50-160.06599 50-26 -0.00963 50-17 0.05644 50-27 0.00379 50-18 0.25525 50-280.00674 50-19 0.01374 50-29 0.00360 -20 0.04345 50-30 0.00575 50-210.00802 50-31 -0.00143 50-22 0.02203 50-32 -0.00817 50-23 0.00356 50-330.00739 50-24 0.01858 50-34 0.02271 50 50-25 -0.00543 50-35 0.0372850-26 0.01094 50-36 0.06442 50-27 0.00452 50-28 0.0075 1 Cornbinor 60-3250-29 0.00422 50-30 -0.00639 50-1 0.01544 1.1675 5031 504 (10002 50-32-0.00906 50 3 00934 50-33 0.00813 50 4 0 03g19 50-34 0.025 17 50.5 g 5350-35 0.04125 59 6 000755 50-36 0.07136 50-7 0.01350 50 3 001013Combinor 60-34 50-9 0.00461 50-10 0.00099 5 50-1 0.01420 1.3156 50-110.03874 50-2 0.00012 50-12 0.04434 50-3 0.08547 50-13 0.053 17 50-40.03542 50-14 0.01631 50-5 0.00822 50-15 0.28436 7 50-6 0.00638 50-160.041 15 50-7 0.01186 50-17 0.087 35 50-8 0.00939 50-18 0. 12108 50-90.00491 50-19 0.00844 50-10 0.00114 25 26 TABLE IIContinued TABLEII-Continued Logarithm Logarithm generator y K generator y K 50420.03799 5045 0.07784 5043 -003377 504 0.03253 50-14 0.04971 50-5 0.007685045 0.05700 50-6 0.00561 50-16 0.07753 504 0.01055 50-17 0.20037 50 80.00826 50-18 0.26342 50-9 0.00417 5049 0.04328 50-10 0.00080 5040-0.03262 5041 0.02252 5041 0.01491 5042 0.03415 5042 0.02250 50430.03656 5043 0.00503 5044 0.04236 5044 0.01973 5045 0.06927 5025 0.0060750-16 -0.06639 50-26 0.01160 5047 0.18672 5047 0.00497 5048 0.1651650-28 0.00789 5049 0.28028 5049 0.00458 5040 -0.01438 50-30 -0.006710.21 0.03082 50-31 0.00190 5042 0.02489 5042 -0.00951 5023 -0.0086350-33 0.00848 5044 -0.02251 50-34 --0.02641 5045 -0.00760 50-35 0.0432450-26 -0.01308 50-36 0.07 5047 0.00605 50-28 -0.00867 Combinor 60-355049. 0.00543 5030 -0.00736 50-1 0.01356 1.3519 5031 0.00232 50-20.00019 50 32 0.01044 50-3 0.08149 50 33 0.00919 5041 0.03390 50-340.02896 50-5 -0.00793 50-35 0.04732 50-6 0.00597 50-36 0.08207 51140.01116 4 50-8 0.00880 Combinor 60-37 50 9 0.00456 5040 0.00103 50 10.01244 1.5271 5041 0.02404 50-2 0.00041 5042 0.03541 -3 0.07437 5043-0.03614 4 0.03128 50 4 0.05021 5115 -0.00747 5045 0.06100 50-6 0.005245046 0.05159 50 7 0.00996 5047 0.30928 50-8 0.00778 5021 -0.103S5 50-90.00385 5049 0.18351 5040 0.00061 50-20 -0.00364 5011 0.02054 50410.02782 2 2 0.03356 50-22 0.02144 50-13 0.03588 50-23 --0.00731 50440.03546 50.24 0.02075 5045 0 .06797 5045 0.00686 50-16 0.03496 50-260.01223 5041 0.04435 50-27 0.00551 50-18 0.28094 5048 (100822 50490.14266 5049 5040 0.14463 50-30 0.00700 5041 0.00328 50 31 -0.00211 5042-0.04007 50-32 0.00994 50-23 -0.00727 50 33 0.00881 5044 -0.02575 50-340.02760 50-25 5045 0.04515 50-26 -0.01430 50-36 0.07824 5027 0.006655048 -0.00926 Combinor 60-36 5049 -0.00594 50-30 0.00781 50-1 0.012981.4036 50-31 0.00257 50-2 0.00027 50-32 0.01104

